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Appendix II.                             867

to o T shew angular velocity of driver A B, taken constant, while
OR to ow shew varying angular velocity of follower CD. For,
work done being equal, area OPT must equal area o R w, which is
only true for circle and ellipse respectively. Thus, area of circle
= ?r (o x)2, and area of ellipse = TT (o R) (o w), which is TT (ox)2 by

P. 577. Rolling Curves.  Other curves may be produced
in the manner described at p. 517. Thus, when a parabola is
rolled on a straight line, its focus describes a catenary^ the curve
in which a chain hangs, and whose equation is

where e= 2718, and m is a constant depending on the depth of
hang. The ordinate is y and the abscissa x. The tractrix or
anti-friction curve, is the involute of the catenary.

P. 528. Tension of Belts. Let the arc 0, Fig. 830, of

the pulley A, be wrapped by a belt, the greatest pull Tn being
balanced by /n+ friction. Considering a small elemental strip
On, the tensions /i and / are balanced by pressure /, and /i= /
c diaram

approximately.    By force diagram

p     chord

-             ,  ,      , n

__     on       and/ = / 0n
t     radius

Frictional resistance on small arc = /
/. //u = p. /0n